Disclosing recurrence properties in fluidized beds

In this paper we shortly investigate the similarity between states for different fluidized bed regimes, which is an essential requirement for the application of recurrence computational fluid dynamics (rCFD) [Lichtenegger and Pirker, Chem. Eng. Sci. 153, 394 (2016)]. Therefore, bubbling and turbulent fluidization regimes are outlined in the frame of different variable-based recurrence/distance norms (rNorm/dNorm). This last is typically used to quantify the degree of flow similarity within the observation time span. The recurrence plots or the so-called recurrence/distance matrices are accordingly constructed upon the rNorm/dNorm values and show no recurrence tendency for the full-resolved turbulent fluidization regimes. It is reported thereabouts that the high fluctuated nature of turbulent fluidization (interphase small scales) absentees the periodicity, and a disclosing procedure for the superstructure (large-scale) dynamics is vitally needed. To that end, a spatial filtering with the idea of the domain decomposition (recurrence island) has been applied to reveal a proper posterior indicator of turbulent fluidization recurrence. The filtering approach handles in increasing the recurrence prominence without changing the system behavior and makes the recurrence more visible. From other perspectives, a turbulence modeling using the coarse-grained approximate deconvolution model–two-fluid model (ADM-TFM) is employed for the turbulent fluidization case. Following its recurrence investigations, the resultant prominence offered by ADM-TFM is in a very comparable aspect to the same grid spatial filtering.

Figure 1

The dMatrix of the dNorm (a) $D^{{\alpha }_s}$ and (b) $D^{{\mathit{\varphi }}_s}$ for the bubbling fluidized bed.

Figure 2

Midwidth planes of ${\alpha }_s$ at two recurrence states, (a) $D_1^{{\alpha }_s}(121,32)=0.428$ [black circle in Fig. 1], and (b) $D_2^{{\alpha }_s}(270,140)=0.52$ [white circle in Fig. 1] in the bubbling fluidized bed.

Figure 3

The rVector profiles taken along the horizontal line $m=30$ on (a) dMatrix $D^{{\alpha }_s}$ in Fig. 1 and $m=20$ on (b) dMatrix $D^{{\mathit{\varphi }}_s}$ in Fig. 1 in the bubbling fluidized bed.

Figure 4

(a) displays the probability density function (PDF) distribution of $\parallel \mathbf{\nabla }{\mathit{\varphi }}_s\parallel$ at different time segments in the bubbling FB, where $\parallel \mathbf{\nabla }{\mathit{\varphi }}_s\parallel$ data are represented on the logarithmic scale after an exponential bin width division. Note that the highest PDF event which corresponds to the gas-filled areas outside the bed, i.e., $\parallel \mathbf{\nabla }{\mathit{\varphi }}_s\parallel =0$, is omitted. (b) represents the mean (off-diagonal) most similarity for ${\alpha }_s$ [Eq. (3)] in the bubbling FB.

Figure 5

The dMatrix of the dNorm $D^{{\alpha }_s}$ in (a) and $D^{{\mathit{\varphi }}_s}$ in (b) for the turbulent fluidized bed.

Figure 6

The rVector profiles taken along the horizontal line $m=60$ on the dMatrix $D^{{\alpha }_s}$ in Fig. 5 and $m=65$ on the rMatrix $D^{{\mathit{\varphi }}_s}$ in Fig. 5 in the turbulent fluidized bed.

Figure 7

(a) displays the PDF distribution of $\parallel \mathbf{\nabla }{\mathit{\varphi }}_s\parallel$ at different time segments in the turbulent FB, where $\parallel \mathbf{\nabla }{\mathit{\varphi }}_s\parallel$ data are represented on the logarithmic scale after an exponential bin width divisions. (b) represents the mean (off-diagonal) most similarity for ${\alpha }_s$ [Eq. (3)] in the turbulent FB.

Figure 8

(a) shows midwidth planes of $\parallel {\mathit{\varphi }}_s\parallel$ (first left), ${\alpha }_s$ (second left), $\parallel \mathbf{\nabla }{\mathit{\varphi }}_s\parallel >50$ (first right), and $\parallel \mathbf{\nabla }{\mathit{\varphi }}_s\parallel <50$ (second right), taken from the turbulent fluidized bed at state $(n=125)$. (b) depicts the dMatrices of $D^{\parallel \mathbf{\nabla }{\mathit{\varphi }}_s\parallel }$ when $\parallel \mathbf{\nabla }{\mathit{\varphi }}_s\parallel$ is not conditioned (top) or conditioned by $\parallel \mathbf{\nabla }{\mathit{\varphi }}_s\parallel >50$ (middle) and $\parallel \mathbf{\nabla }{\mathit{\varphi }}_s\parallel <50$ (bottom) for the same turbulent FB.

Figure 9

Midwidth plane of ${\alpha }_s$ taken from the turbulent fluidized bed: (a) Initial-state $n=14$. (b) State $n=125$. (c) Filtered $n=125$ using box filter $\overline{\mathrm{\Delta }}=12\mathrm{\Delta }{\mathit{r}}_i$. (d) Filtered $n=125$ using elliptic filter ${\mathcal{F}}^{\left(128\right)}$. (e) Filtered $n=125$ using $10\times$ binning procedure. (f) presents the variance $\langle {\alpha }_s^{\prime 2}\rangle$ as an indicator for the less homogeneous area, i.e., rIsland (the gray line frame).

Figure 10

dMatrices of $D^{{\alpha }_s}$ in (a) and its outcomes applying filtered fields $D^{\overline{{\alpha }_s}}$ using the box filter $\overline{\mathrm{\Delta }}=12\mathrm{\Delta }{\mathit{r}}_i$ (b), the $10\times$ binning procedure (c), and the elliptic filter ${\mathcal{F}}^{\left(128\right)}$ (d). The normalization maximals, i.e., denominator in Eq. (1) scaled by the volume, are as follows: (a) 0.05, (b) 0.03, (c) 0.03, and (d) 0.016.

Figure 11

dMatrices of the rIsland displayed in Fig. 9 (gray line frame) for $D^{{\alpha }_s}$ in (a) and its outcomes applying filtered fields $D^{\overline{{\alpha }_s}}$, the $10\times$ binning procedure (b), and the elliptic filter ${\mathcal{F}}^{\left(128\right)}$ (c). The normalization maximals, i.e., denominator in Eq. (1) scaled by the volume, are as follows: (a) 0.035, (b) 0.025, and (c) 0.014.

Figure 12

The rVector profiles taken along the horizontal line $m=20$ on the dMatrices in Figs. 10 and 11 and normalized by its own rPlateau.

Figure 13

(a) represents the dMatrices of $D^{\overline{{\mathit{\varphi }}_s}}$ (top), $D^{\overline{{\alpha }_s}}$ (middle), and the rVector along $m=20$ on $D^{\overline{{\alpha }_s}}$ dMatrix, normalized by its rPlateau (bottom), for the coarse-grained (ADM-TFM) turbulent fluidized bed. (b) displays midwidth planes of one recurrent point $D^{\overline{{\alpha }_s}}(42,3)$ (black circle) in (a) (middle). The normalization maximals, i.e., denominator in Eq. (1) scaled by the volume, are as follows: (a) top 0.07 and (a) middle 0.035.

Figure 14

jVectors extracted from the dMatrices in Fig. 10.

Figure 15

Two-point correlation profiles of ${\alpha }_s$ starting from the midlength plane towards the walls and extracted at different heights $\left\{0.05\text{:}0.8\right\}$ inside the turbulent FB.